# Largest trapezoid that can be inscribed in a semicircle

Given a semicircle of radius **r**, the task is to find the largest trapezoid that can be inscribed in the semicircle, with base lying on the diameter.

**Examples:**

Input: r = 5 Output: 32.476 Input: r = 8 Output: 83.1384

**Approach**: Let **r** be the radius of the semicircle, **x** be the lower edge of the trapezoid, and **y** the upper edge, & **h** be the height of the trapezoid.

Now from the figure,

r^2 = h^2 + (y/2)^2

or, 4r^2 = 4h^2 + y^2

y^2 = 4r^2 – 4h^2

y = 2√(r^2 – h^2)

We know, Area of Trapezoid, A = (x + y)*h/2

So, A = hr + h√(r^2 – h^2)

taking the derivative of this area function with respect to h, (noting that r is a constant since we are given the semicircle of radius r to start with)

dA/dh = r + √(r^2 – h^2) – h^2/√(r^2 – h^2)

To find the critical points we set the derivative equal to zero and solve for h, we get

h = √3/2 * r

So, x = 2 * r & y = r

So,

A = (3 * √3 * r^2)/4

**Below is the implementation of above approach**:

## C++

`// C++ Program to find the biggest trapezoid ` `// which can be inscribed within the semicircle ` `#include <bits/stdc++.h> ` `using` `namespace` `std; ` ` ` `// Function to find the area ` `// of the biggest trapezoid ` `float` `trapezoidarea(` `float` `r) ` `{ ` ` ` ` ` `// the radius cannot be negative ` ` ` `if` `(r < 0) ` ` ` `return` `-1; ` ` ` ` ` `// area of the trapezoid ` ` ` `float` `a = (3 * ` `sqrt` `(3) * ` `pow` `(r, 2)) / 4; ` ` ` ` ` `return` `a; ` `} ` ` ` `// Driver code ` `int` `main() ` `{ ` ` ` `float` `r = 5; ` ` ` `cout << trapezoidarea(r) << endl; ` ` ` `return` `0; ` `} ` |

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## Java

`// Java Program to find the biggest trapezoid ` `// which can be inscribed within the semicircle ` ` ` `import` `java.util.*; ` `import` `java.lang.*; ` `import` `java.io.*; ` ` ` `class` `GFG{ ` `// Function to find the area ` `// of the biggest trapezoid ` `static` `float` `trapezoidarea(` `float` `r) ` `{ ` ` ` ` ` `// the radius cannot be negative ` ` ` `if` `(r < ` `0` `) ` ` ` `return` `-` `1` `; ` ` ` ` ` `// area of the trapezoid ` ` ` `float` `a = (` `3` `* (` `float` `)Math.sqrt(` `3` `) ` ` ` `* (` `float` `)Math.pow(r, ` `2` `)) / ` `4` `; ` ` ` ` ` `return` `a; ` `} ` ` ` `// Driver code ` `public` `static` `void` `main(String args[]) ` `{ ` ` ` `float` `r = ` `5` `; ` ` ` `System.out.printf(` `"%.3f"` `,trapezoidarea(r)); ` `} ` `} ` |

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## Python 3

`# Python 3 Program to find the biggest trapezoid ` `# which can be inscribed within the semicircle ` ` ` `# from math import everything ` `from` `math ` `import` `*` ` ` `# Function to find the area ` `# of the biggest trapezoid ` `def` `trapezoidarea(r) : ` ` ` ` ` `# the radius cannot be negative ` ` ` `if` `r < ` `0` `: ` ` ` `return` `-` `1` ` ` ` ` `# area of the trapezoid ` ` ` `a ` `=` `(` `3` `*` `sqrt(` `3` `) ` `*` `pow` `(r,` `2` `)) ` `/` `4` ` ` ` ` `return` `a ` ` ` ` ` `# Driver code ` `if` `__name__ ` `=` `=` `"__main__"` `: ` ` ` ` ` `r ` `=` `5` ` ` ` ` `print` `(` `round` `(trapezoidarea(r),` `3` `)) ` ` ` ` ` `# This code is contributed by ANKITRAI1 ` |

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## C#

`// C# Program to find the biggest ` `// trapezoid which can be inscribed ` `// within the semicircle ` `using` `System; ` ` ` `class` `GFG ` `{ ` `// Function to find the area ` `// of the biggest trapezoid ` `static` `float` `trapezoidarea(` `float` `r) ` `{ ` ` ` ` ` `// the radius cannot be negative ` ` ` `if` `(r < 0) ` ` ` `return` `-1; ` ` ` ` ` `// area of the trapezoid ` ` ` `float` `a = (3 * (` `float` `)Math.Sqrt(3) * ` ` ` `(` `float` `)Math.Pow(r, 2)) / 4; ` ` ` ` ` `return` `a; ` `} ` ` ` `// Driver code ` `public` `static` `void` `Main() ` `{ ` ` ` `float` `r = 5; ` ` ` `Console.WriteLine(` `""` `+ trapezoidarea(r)); ` `} ` `} ` ` ` `// This code is contributed ` `// by inder_verma ` |

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## PHP

`<?php ` `// PHP Program to find the biggest ` `// trapezoid which can be inscribed ` `// within the semicircle ` ` ` `// Function to find the area ` `// of the biggest trapezoid ` `function` `trapezoidarea(` `$r` `) ` `{ ` ` ` ` ` `// the radius cannot be negative ` ` ` `if` `(` `$r` `< 0) ` ` ` `return` `-1; ` ` ` ` ` `// area of the trapezoid ` ` ` `$a` `= (3 * sqrt(3) * pow(` `$r` `, 2)) / 4; ` ` ` ` ` `return` `$a` `; ` `} ` ` ` `// Driver code ` `$r` `= 5; ` `echo` `trapezoidarea(` `$r` `).` `"\n"` `; ` ` ` `// This code is contributed ` `// by ChitraNayal ` `?> ` |

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**Output:**

32.476

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